420 , Mr. Bowditch on the motion ofa pendulum pee 
calf aab cosine (at+c) a 
dy” a’a’b’ cosine (atc) - rap which by §5 is eg? to = sii ‘ 
axa 
hence the subtangent “= _ becomes 
struction. Take on ag axis NS (Plate 3, Fig. 2 or 4) GI=GL= 
“.GS, and join KI, AL, DI, CL, these will be respectively the 
tangents of the curve at the points K, A, D, C. If we substitute the 
values, sine (at-+c) =o, sine (a’t+c’) =o, in the equations (C) we shall 
find the velocities in the directions of the axes a, y, are equal to 9, 
. 6, which affords this con- 
consequently the pendulum must be at rest when it is at —_ * the 
points A, D, K or C. 
9. In the case of the last article where 4’=o, the vibrations must ~ 
be performed in the plane za, exactly like those of a pendulum of the 
length r suspended from a fixed point C. (Plate 3. Fig. 1.) Be- 
cause the values a=0 cosine (at+c) given by the first of the equations 
(B) is not altered by supposing AB and x’ to be decreased till they 
become 9, and this value of a shows that one vibration 1s performed 
while 2 changes from 6 to —, consequently the arch a¢ must ig 
180° in one vibration. 
_ In like manner when 6=0, the body must move. in the plane 9 
and one vibration will be performed while y=d' cosine (a't+¢) changes 
~ from 8’ to —Z' ; consequently, in the time of one vibration, the 
@'t will increase 180°, and the body will vibrate like a simple pend 
lum of the length r+’, suspended from the point G. For aa’ being 
——— the value a it would not be changed by decreasing r till it ‘be. 
came 0, provided 7 were increased by an equal quantity The same 
would take place if r were decreased till it became 0, 7’ being 1 
by an equal quantity, so as to preserve the same value to r+r and a. 
